implement numerical evaluation of erf at complex arguments via mpmath algorithm

[williamstein]

When implemented, this would work:

sage: erf(2.0, algorithm='mpmath')
...

---

Apply patch trac_1173_complex_erf_v3.patch attached to this ticket to the Sage library.

CC: @benjaminfjones

Component: calculus

Keywords: sd35.5

Reviewer: Karl-Dieter Crisman

Issue created by migration from https://trac.sagemath.org/ticket/1173

[williamstein]

Description changed:

--- 
+++ 
@@ -1,5 +1,7 @@
 When implemented, this would work:
 
+```
 sage: a = sqrt(pi)*I*erf(2*I)/2
 sage: CC(a)
 ...
+```

[williamstein]

comment:1

See also
#1174

[boothby]

comment:2

The following has a GPL'd implementation in c:

http://velveeta.che.wisc.edu/octave/lists/archive/octave-sources.1998/msg00032.html

[williamstein]

comment:3

This looks like the wya to go:

Josh Kantor:

scipy has an error function that takes complex arguments

sage:  import numpy, scipy
sage:  from scipy import special
sage:  j=numpy.complex(0,1)
sage: -j*float(sqrt(pi))*special.erf(2*j)/2
(16.45262776550727+0j)

Unfortunately numpy and sage's complex numbers are not compatible yet.

[kcrisman]

comment:4

Numpy and Sage numbers now seem to work well, at least to some extent:

sage: import numpy
sage: CC(numpy.complex(0,1))
1.00000000000000*I
sage: CC(numpy.complex(1,1))
1.00000000000000 + 1.00000000000000*I
sage: import scipy
sage: CC(scipy.special.erf(numpy.complex(1,1)))
1.31615128169795 + 0.190453469237835*I

Is it time to wrap this now, two years after opening?

[aghitza]

comment:5

This can also be done by mpmath in arbitrary precision, see

sage: import mpmath
sage: mpmath.erf?

[kcrisman]

comment:6

Replying to @aghitza:

This can also be done by mpmath in arbitrary precision, see

sage: import mpmath
sage: mpmath.erf?

And erf already has an _eval_f method, so maybe this could be changed to use mpmath? Even a loss in speed would be worth having the complex values. This could apply to error_fcn/erfc and others as well.

[kcrisman]

comment:7

Update - yes, we should definitely do this because of the complex values - just had a support request about not being able to do

sage: integrate(sin(x^2),(x,2,6))

and then use n() because of this (partly). See also #9044.

[sagetrac-dsm]

comment:8

Okay, here's v0.prealpha0.0_1a of a patch which uses mpmath to get complex and arbitrary-precision behaviour for erf. (If it works out we can do error_fcn the same way, as noted by kcrisman).

Unfortunately it does introduce a speed regression:

# 4.6.1, via pari
sage: timeit('float(erf(0))',number=1000) # must be float because 4.6.1 doesn't evaluate
1000 loops, best of 3: 109 µs per loop
sage: timeit('erf(0.1)',number=1000)
1000 loops, best of 3: 98.4 µs per loop
sage: timeit('erf(0.99)',number=1000)
1000 loops, best of 3: 98.5 µs per loop

# 4.6.2rc0 with patch
sage: timeit('erf(0)',number=1000)
1000 loops, best of 3: 68.3 µs per loop
sage: timeit('erf(0.1)',number=1000)
1000 loops, best of 3: 154 µs per loop
sage: timeit('erf(0.99)',number=1000)
1000 loops, best of 3: 165 µs per loop

I attempted to fix this by using the old approach for the default precision, but it usually wound up being more expensive to do the tests. Someone with better speed-fu is encouraged to look at it. I haven't finished a long testall yet, so there's probably something somewhere which breaks, but here's the first attempt.

[burcin]

Author: D. S. McNeil